Introduces the idea of “counting board” - behaves like an “augmented matrix”.
Input System of Equations:
(\begin{aligned} 2x + 3y - z = 5 \ - x + 4y + 2z = 6 \ 3x - y + z = 4 \end{aligned})
Augmented Matrix:
( \left[ \begin{array}{ccc|c} 2 & 3 & -1 & 5 \ -1 & 4 & 2 & 6 \ 3 & -1 & 1 & 4 \end{array} \right] )
He uses determinants to solve systems with unique solutions based on his system.
A detailed account of how Gauss’s method came to be - is explained in this page: How ordinary elimination became Gaussian elimination. Interestingly - there is a Newton’s notebook connection to the topic, and a lot of backstory behind how the method got published.
In the following page from Sylvester’s…
Introduces the idea of “counting board” - behaves like an “augmented matrix”.
Input System of Equations:
(\begin{aligned} 2x + 3y - z = 5 \ - x + 4y + 2z = 6 \ 3x - y + z = 4 \end{aligned})
Augmented Matrix:
( \left[ \begin{array}{ccc|c} 2 & 3 & -1 & 5 \ -1 & 4 & 2 & 6 \ 3 & -1 & 1 & 4 \end{array} \right] )
He uses determinants to solve systems with unique solutions based on his system.
A detailed account of how Gauss’s method came to be - is explained in this page: How ordinary elimination became Gaussian elimination. Interestingly - there is a Newton’s notebook connection to the topic, and a lot of backstory behind how the method got published.
In the following page from Sylvester’s “Additions to the Articles on a New Class of Theorems” (link) - we can find the first reference to the word “Matrix”. Matrix is derived from “mater” - which in latin stands for mother. He visualized matrix as mother’s womb, which gives birth to many determinants - smaller matrices derived from the original by removing rows and columns.
In this work, he comes up with the Ax = b notation for representing system of equations even today
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